Philip Kokic Mathematics, Informatics and Statistics CSIRO Location: Carslaw 173 Time: 2pm Friday, August 6, 2010 Title: Multivariate quantiles, expectiles and M-quantiles Abstract: Ordinary quantiles and expectiles are special cases of univariate M-quantiles, which can be defined either in terms of the minimum of a specific loss function or the solution to the corresponding estimating equation. In this presentation we show how M-quantiles can be extended to a multivariate situation by generalising the estimating equation approach. Some of the features and properties of Multivariate M-quantiles are explored using a variety of simulated data sets. In particular, it is illustrated how, to a certain extent, they are able to capture non-linear relationships between variables. For data of more than 1 dimension they are unique and can also be computed efficiently even for high dimensional data. Unlike alternatives based on multivariate density estimates they do not suffer from ’the curse of dimensionality’. Multivariate outliers present some interesting statistical issues; for example, an outlier may or may not become an influential observation depending on the type of analysis performed. In this presentation we also describe how M-quantiles need to be modified for multivariate outlier detection. To this end a semi-parametric form of multivariate expectiles is described, and the approach is illustrated using UK annual business survey data.